Would either $L_1$ or $L_2$ regularisation lower the MSE on the training and test data?
The answer is largely the same whether we consider $\ell_1$ or $\ell_2$ regularisation, so I will just speak generally about regularisation.
Mean square error for training data
Given some training data $\{(x_i, y_i)\}_{i = 1}^n$, a linear regression line $Y = aX + b$ fit using the least squares method looks for coefficients that minimise the sum of squares, i.e. they are the minimisers given by
$$ \mathrm{arg\,min}_{a, b} \sum_{i = 1}^n \left(y_i - (ax_i + b)\right)^2.$$
This gives the same coefficients as minimising the mean square error
$$ \mathrm{MSE}\left((x_1, y_1), \dots, (x_n, y_n)\right) = \frac{1}{n} \sum_{i = 1}^n \left(y_i - (ax_i + b)\right)^2.$$
So, by definition, the coefficients $(a, b)$ minimise the MSE on the training data. Any regularisation will only increase the MSE on the training data.
Generalisation performance
The main point of regularisation is to prevent overfitting on the data and improve the generalisation performance (i.e. on the test set).
With an appropriate parameter for regularisation, you may obtain a smaller MSE on the test set. This depends on your dataset and the parameters you choose: strong regularisation may lead to underfitting, whereas weak regularisation might not make much difference to the coefficients that you fit.